Median Balls: An Extension of the Interquantile Intervals to Multivariate Distributions

For a probability distribution on a Banach space, we introduce a family of central balls, indexed by their radius, using a proximity criterion close to those defining the spatial median. It is shown that these balls possess robustness and equivariance properties similar to those of the spatial median. They provide a multivariate generalization of the real interquantile intervals and can be interpreted as trimmed regions.